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http://www.staff.science.uu.nl/~delde102/StudyAdvice_MPOC-master.htm
Second year master programme
Simulation of Ocean, Atmosphere and Climate (7.5 ec)Making, Analysing and Interpreting Observations (7.5 ec)
Thesis Research Project (45 ec) (periods 2-4)
http://www.staff.science.uu.nl/~delde102/SOAC.htm
Simulation of Ocean, Atmosphere and Climate (SOAC)
week 36: lectures, invited talks & exercises
week: 37, 38, 39: project under supervision of IMAU-staf-member
(in couples)
week 40Thu. 2 Oct.: oral presentation of project
write a report (<3000 words)
Schedule: SOAC 2014 (http://www.staff.science.uu.nl/~delde102/SOAC.htm) Monday 1 September (MIN 025) 09:15-9:30: Aarnout van Delden: Introduction to the “research” year of the master program 09:30-10:15: Aarnout van Delden: Numerical Fluid Dynamics 1 10:30-12:15: Lars Tijssen: Introduction to Python 12:15-13:00: Aarnout van Delden: Introduction to Exercises 1(weather and climate of a simple recursive model) and 2 (Lagrangian model of the vertical motion of a buoyant fluid parcel) Afternoon: Working on exercise 1 and 2 Tuesday 2 September (MIN 205) Hand in answers to exercise 1 (individually) 09:15-09:45: Aarnout van Delden: Discussion of exercise 1 09:45-10:30: Aarnout van Delden: Numerical Fluid Dynamics 2 Rest of the day: Working on exercise 2 Wednesday 3 September (MIN 205) 09:15-10:00: Aarnout van Delden: Numerical Fluid Dynamics 3 10:15-11:00: Michael Kliphuis: Computer hardware and climate models 11:00-12:45 Sander Tijm: Hydrostatic and non-hydrostatic limited area models for weather prediction 13:30-14:15: Aarnout van Delden: Introduction to Exercise 3 (Solving the advection equation with different numerical schemes) Afternoon: Working on exercise 2 and 3 Thursday 4 September (MIN 205) Hand in answers to exercise 2 (individually) 09:15-10:15: Dewi Le Bars, Wim Ridderinkhof/Niels Alebregtse, Carles Penades/ Huib de Swart, Willem-Jan van den Berg, Anna von der Heydt, Rianne Giesen, Claudia Wieners and Aarnout van Delden: Overview of the projects 10:30-10:50: Aarnout van Delden: Discussion of exercise 2 10:50-11:15: Lars Tijssen: presentation (an extension of exercise 2) 11:30-13:15 Jordi Vila: Large Eddy Simulation Afternoon: Working on exercise 3 Friday 5 September (MIN 205) Hand in answers to exercise 3 (individually) 9:15-10:00: Aarnout van Delden: Discussion of exercise 3 10:15-11:15: Rein Haarsma: The atmosphere in EC-Earth 11:30-13:15: Dewi Le Bars: The ocean in CESM-climate model Finally: Who is who with the projects (couples)? Thursday 2 October (HFG 611AB) 9:15-12:00: Presentations of the results from the projects
Programming languages
FortranC/C++Pascal
MATLABPython
…
Programming languages
Advantages of Python over MATLAB:1) Python code is more compact and readable than Matlab code.
2) The Python world is free and open (in several senses).
3) Like C/C++, Java, Perl, and most other programming languages other than Matlab, Python conforms to certain de facto standards, including zero-based indexing and the use of square brackets rather than parentheses for indexing..
4) Python makes it easy to maintain multiple versions of shared libraries
5) Python offers more choice in graphics packages and toolsets
Pythonhttps://www.python.org/
http://www.numpy.org/
http://matplotlib.org/index.html
http://www.johnny-lin.com/pyintro/
http://www.staff.science.uu.nl/~delde102/StudyAdvice_MPOC-master.htm
Master thesisFrom November/December you will need to find a
thesis project. Decide what you find interesting. Talk to potential thesis supervisors. It is also possible to do a
thesis project at KNMI, NIOZ or any other (foreign)university. However, next to the daily
supervisor at the other institute, you must always have a staff member of IMAU as second supervisor.
You must make clear arrangements about supervision and about what is expected of you. Independence
and originality are very much appreciated! At the start you have to fill in a “research application form”
Introduction to numerical fluid dynamics for geophysical flows
Anna von der HeydtAarnout van Delden
a.j.vandelden@uu.nl BBL 615
This visualization shows early test renderings of a global computational model of Earth's atmosphere based on data from NASA's Goddard Earth Observing System Model, Version 5 (GEOS-5). This particular run, called 7km GEOS-5 Nature Run (7km-G5NR), was run on a supercomputer, spanned 2 years of simulation time at 30 minute intervals, and produced Petabytes of output. The model uses a 7.5 km cube-sphere parameterization. Geographic coordinate output volumes from the model are 5760 x 2881 x 72 voxels per time step. For each voxel numerous physical parameters are available such as temperature, wind speed and direction, pressure, humidity, etc. This visualziation uses a combination of the CLOUD and TAUIR parameters.The visualization spans a little more than 7 days of simulation time which is 354 time steps. The time period was chosen because a simulated category-4 typhoon developed off the coast of China. The frames were rendered using Renderman. Brickmap volumes generated for each time step are about 2.6 Gigabytes. This short visualization referenced nearly a terabyte of brickmap files. The 7 day period is repeated several times during the course of the visualization.This animation was presented at SIGGRAPH 2014 during the Dailies session. (http://svs.gsfc.nasa.gov/vis/a000000/a004100/a004180/)
http://www.gfdl.noaa.gov/visualization
climate modellinghttps://www.youtube.com/watch?v=ADf8-rmEtNg
High resolution climate model outputhttps://www.youtube.com/watch?v=Cxsg7uvVSBE
Uncertainty in climate modelshttps://www.youtube.com/watch?v=4AjCeXl5tE0
Partial differential equationshttps://www.youtube.com/watch?v=fYVMmEykiMw
Cloud model outputhttps://www.youtube.com/watch?v=mlvLX7YvI88
Ocean modellinghttps://www.youtube.com/watch?v=B-TSwthjPYE
GFDL visualizationhttp://www.gfdl.noaa.gov/visualization
Animations and lectures
Earth system models
NCAR Community Earth System Model (CESM)
Atmosphere grid box: wind vectors, humidity, clouds, temperature, chemical species.
Surface: ground temperature, water, energy, momentum, CO2 fluxes.
Ocean grid box: current vectors, temperature, salinity.
General circulation models (GCMs)
Resolution in IPCC simulations(IPCC 1990)
(IPCC 1996)
(IPCC 2001)
(IPCC 2008)
Spin up• Spin up time =
integration time, the model needs to reach a (statistical) equilibrium.
• Depends on the slowest component of the modelled climate system:
• Atmosphere ~ 15 years.• Ocean ~ 3000 years.• Ice sheets ~ even longer.
How to build a numerical model for a fluid system?���
(air or water)Recommended books:
Benoit Cushman-Roisin, Jean-Marie Beckers, Introduction to Geophysical Fluid Dynamics - Physical and Numerical Aspects. 2nd Edition, Academic Press (2011) (chapters 5 & 6)
Dale R. Durran, Numerical Methods for Fluid Dynamics – With applications to geophysics, 2nd Edition, Springer (2010)
SHALLOW-WATER EQN’S(the “e-coli” of Geophysical Fluid Dynamics)
characteristics
solution
what about boundary conditions?
ADVECTION EQUATION
FINITE DIFFERENCE
stencil
forward
backward
central
ACCURACYTaylor expansion
truncation error
the lowest order of in the truncation error determines the accuracy
finite difference approximation exact
ACCURACYtruncation error
the lowest order of in the truncation error determines the accuracy
Lecture 2
Grading
Exercises 1-3 + attendance first week: 20% of grade
Oral presentation: 40%Written report: 40%
ADVECTION EQUATION
truncation errorcomputer sets this to zero
notation:
this scheme is first-order accurate in time and space
ADVECTION EQUATION
truncation error
numerical diffusion numerical dispersion
∂ 2u∂t2
= −c ∂∂t
∂u∂ x
= −c ∂∂ x
∂u∂t
= −c2 ∂2u
∂ x2
STABILITYVon Neumann’s Method
insert Fourier series to represent the discretized solution
at
at
note that
amplification factor
stable if
(page 96 Durran)
Analysis is restricted to linear equations, implying that amplification does not vary from time step to time step
STABILITYinsert
rewriting
for every
most unstable modealways unstable
euler forward / downwind
STABILITYinsert
rewriting
for every
most unstable mode
euler forward / upwind
stable if Courant Frederich Lewy condition
CFL CONDITION
euler forward / upwindeuler forward / downwind
always unstable Courant Frederich Lewy condition
unstablestable
IMPLICIT SCHEMEinsert
euler backward / upwind
unconditionally stable!
advantage: we can take large time steps
but 1: accuracy determines time step
but 2 : implicit schemes are harder / impossible to solve
in general form Euler forward Euler backward
DIFFUSION EQUATION
The finite difference approximation for the second derivative
DIFFUSION EQUATION
euler forward / central difference
accuracy and
stability
condition
MATSUNO SCHEMETime differencing:
€
C * jn+1 −Cj
n
Δt= u0
∂C∂x% & '
( ) * j
n
€
Cjn+1 −C j
n
Δt= u0
∂C *∂x
% & '
( ) * j
n+1
Step 1
Step 2
€
∂C∂t
= −u0∂C∂x
Lax-Wendroff schemeA well-known finite difference scheme:
Taylor series:
Since€
Cjn+1 =Cj
n +Δt ∂C∂t
$ % &
' ( ) j
n+Δt2 ∂
2C∂t2$
% &
'
( ) j
n
+ ...
€
∂2C∂t2
= u02 ∂
2C∂x2
€
Cjn+1 =Cj
n − u0Δt∂C∂x% & '
( ) * j
n+u02Δt2
2∂2C∂x2%
& '
(
) * j
n
+ ...
€
∂C∂t
= −u0∂C∂x
Lax-Wendroff scheme
€
Cjn+1 =Cj
n − u0Δt∂C∂x% & '
( ) * j
n+u02Δt2
2∂2C∂x2%
& '
(
) * j
n
+ ...
which becomes:
€
Cjn+1 −Cj
n
Δt≈ −u0
Cj+1n −Cj−1
n
2Δx
%
& ' '
(
) * * +
u02Δt2
Cj+1n − 2Cj
n +Cj−1n
2Δx
%
& ' '
(
) * *
€
∂C∂t
= −u0∂C∂x
central difference central difference
Spectral method
€
x j = jΔx, j =1,2,...,N, where NΔx = L.
Set of uniformly spaced grid points in one dimension
The Fourier series of C, whoose values are given only at N grid points, requires N Fourier coefficients. The Fourier expansion of C is
€
C x j( ) = Ck exp 2πikx j /L( )k=1
N∑ = Ck exp 2πikjΔx / NΔx( )( )
k=1
N∑ = Ck exp 2πikj /N( )
k=1
N∑
Ck are the Fourier coefficients. The inverse of this equation yields the complex Fourier coefficients from the grid point values:
€
Ck =1N
C x j( )j=1
N∑ exp −2πikj /N( ) (2)
Spectral methodFor the periodic domain, L, the time-dependent distribution of C can be expressed as
Substituting this equation into the linear advection equation (1) yields N independent first order ordinary differential equations :
At initial time, t=0, we need to compute the complex Fourier coefficients from eq. 2 and then integrate the system (3) in time, implying that we need only to numerically approximate the time derivative. Note that since the Fourier coefficients are complex, (3) represents a system of 2N equations
€
C x,t( ) = Ck t( )exp 2πikx /L( )k=1
N∑
€
dCkdt
= −2πiu0kL
Ck for k =1,2,...,N
€
∂C∂t
= −u0∂C∂x
(1)
(3)
Lecture 3
1D SHALLOW-WATER EQN’S
assume wave solution
dispersion relation
phase velocity
group velocity
WAVE ON A GRID: aliasing
long wave
short wavelimit
aliasing
1D SHALLOW-WATER EQN’S
only spatial discretization
assume wave solution
dispersion relation
phase velocity
group velocity
long wave short wave limit
(linear, non-rotational, inviscid)
�
cg* =
dωdk
= ± gH cos(kΔx)
LEAP-FROG SCHEME
centered in time & centered in spacestencil
grid waves2 decoupled grids
u,h
u,h
u,h
u,h u,h u,h u,h
u,h u,h u,h
u,h u,h
u,h
u,h
u,h
u,h
u,h
u,h u,h u,h
u,h
u,h
u,h
u,h
u,hu,h
u,h
LEAP-FROG SCHEME
centered in time & centered in spacestencil
no grid wavesstaggered grid
h h
u u u u
u u u
h h
u
h
h
h
h
u u u
h
u
h
hh
hh h
two times faster (but also coarser)
Dispersion relation• Numerical phase speeds
�
cu* =
ωk
=c
kΔxsin(kΔx)
�
cs* =
ωk
=2c
kΔxsin(
kΔx2
)
u = unstaggered grid s = staggered grid
2D SHALLOW-WATER EQN’S(linear, inviscid)
assume wave solution
dispersion relation
spatial discretization
again: short waves have too small phase velocity and a group velocity in the wrong direction
2D SHALLOW-WATER EQN’S(linear, inviscid)
assume wave solution
dispersion relation
short waves are gravity waves
long waves are inertial oscillations
STAGGERED GRIDSArakawa Aunstaggered
Arakawa B Arakawa C
h hh
v
interpolation
u* u*
v*
v*
uh
STAGGERED GRIDSArakawa Aunstaggered
Arakawa B Arakawa C
h
u
v
interpolation
u u uh* h*
v*
STAGGERED GRIDS
medium
coarse
fineArakawa B grid
Arakawa C - grid
True solution
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